Theorem int0 4967

Description: The intersection of the empty set is the universal class. Exercise 2 of [TakeutiZaring] p. 44. (Contributed by NM, 18-Aug-1993.) (Proof shortened by JJ, 26-Jul-2021.)

Assertion

Ref	Expression
int0	⊢ ∩ ∅ = V

Proof of Theorem int0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4513	. . . 4 ⊢ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥
2		vex 3479	. . . . 5 ⊢ 𝑦 ∈ V
3	2	elint2 4958	. . . 4 ⊢ (𝑦 ∈ ∩ ∅ ↔ ∀𝑥 ∈ ∅ 𝑦 ∈ 𝑥)
4	1, 3	mpbir 230	. . 3 ⊢ 𝑦 ∈ ∩ ∅
5	4, 2	2th 264	. 2 ⊢ (𝑦 ∈ ∩ ∅ ↔ 𝑦 ∈ V)
6	5	eqriv 2730	1 ⊢ ∩ ∅ = V

Syntax hints:   = wceq 1542   ∈ wcel 2107  ∀wral 3062  Vcvv 3475  ∅c0 4323  ∩ cint 4951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-v 3477  df-dif 3952  df-nul 4324  df-int 4952

This theorem is referenced by:  unissint  4977  uniintsn  4992  rint0  4995  intex  5338  intnex  5339  oev2  8523  fiint  9324  elfi2  9409  fi0  9415  cardmin2  9994  00lsp  20592  cmpfi  22912  ptbasfi  23085  fbssint  23342  fclscmp  23534  zarcmplem  32861  rankeq1o  35143  bj-0int  35982  heibor1lem  36677  ipoglb0  47619  mreclat  47622